# NAG Library Routine Document

## 1Purpose

f07jhf (dptrfs) computes error bounds and refines the solution to a real system of linear equations $AX=B$, where $A$ is an $n$ by $n$ symmetric positive definite tridiagonal matrix and $X$ and $B$ are $n$ by $r$ matrices, using the modified Cholesky factorization returned by f07jdf (dpttrf) and an initial solution returned by f07jef (dpttrs). Iterative refinement is used to reduce the backward error as much as possible.

## 2Specification

Fortran Interface
 Subroutine f07jhf ( n, nrhs, d, e, df, ef, b, ldb, x, ldx, ferr, berr, work, info)
 Integer, Intent (In) :: n, nrhs, ldb, ldx Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (In) :: d(*), e(*), df(*), ef(*), b(ldb,*) Real (Kind=nag_wp), Intent (Inout) :: x(ldx,*) Real (Kind=nag_wp), Intent (Out) :: ferr(nrhs), berr(nrhs), work(2*n)
#include nagmk26.h
 void f07jhf_ (const Integer *n, const Integer *nrhs, const double d[], const double e[], const double df[], const double ef[], const double b[], const Integer *ldb, double x[], const Integer *ldx, double ferr[], double berr[], double work[], Integer *info)
The routine may be called by its LAPACK name dptrfs.

## 3Description

f07jhf (dptrfs) should normally be preceded by calls to f07jdf (dpttrf) and f07jef (dpttrs). f07jdf (dpttrf) computes a modified Cholesky factorization of the matrix $A$ as
 $A=LDLT ,$
where $L$ is a unit lower bidiagonal matrix and $D$ is a diagonal matrix, with positive diagonal elements. f07jef (dpttrs) then utilizes the factorization to compute a solution, $\stackrel{^}{X}$, to the required equations. Letting $\stackrel{^}{x}$ denote a column of $\stackrel{^}{X}$, f07jhf (dptrfs) computes a component-wise backward error, $\beta$, the smallest relative perturbation in each element of $A$ and $b$ such that $\stackrel{^}{x}$ is the exact solution of a perturbed system
 $A+E x^ = b + f , with eij ≤ β aij , and fj ≤ β bj .$
The routine also estimates a bound for the component-wise forward error in the computed solution defined by $\mathrm{max}\left|{x}_{i}-\stackrel{^}{{x}_{i}}\right|/\mathrm{max}\left|\stackrel{^}{{x}_{i}}\right|$, where $x$ is the corresponding column of the exact solution, $X$.
Note that the modified Cholesky factorization of $A$ can also be expressed as
 $A=UTDU ,$
where $U$ is unit upper bidiagonal.

## 4References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
3:     $\mathbf{d}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: must contain the $n$ diagonal elements of the matrix of $A$.
4:     $\mathbf{e}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ subdiagonal elements of the matrix $A$.
5:     $\mathbf{df}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array df must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: must contain the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
6:     $\mathbf{ef}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array ef must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: must contain the $\left(n-1\right)$ subdiagonal elements of the unit bidiagonal matrix $L$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
7:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n$ by $r$ matrix of right-hand sides $B$.
8:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07jhf (dptrfs) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:     $\mathbf{x}\left({\mathbf{ldx}},*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n$ by $r$ initial solution matrix $X$.
On exit: the $n$ by $r$ refined solution matrix $X$.
10:   $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07jhf (dptrfs) is called.
Constraint: ${\mathbf{ldx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
11:   $\mathbf{ferr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: estimate of the forward error bound for each computed solution vector, such that ${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{\stackrel{^}{x}}_{j}‖}_{\infty }\le {\mathbf{ferr}}\left(j\right)$, where ${\stackrel{^}{x}}_{j}$ is the $j$th column of the computed solution returned in the array x and ${x}_{j}$ is the corresponding column of the exact solution $X$. The estimate is almost always a slight overestimate of the true error.
12:   $\mathbf{berr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: estimate of the component-wise relative backward error of each computed solution vector ${\stackrel{^}{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\stackrel{^}{x}}_{j}$ an exact solution).
13:   $\mathbf{work}\left(2×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
14:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

## 7Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E∞=OεA∞$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^ - x ∞ x∞ ≤ κA E∞ A∞ ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{\infty }{‖A‖}_{\infty }$, the condition number of $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Routine f07jgf (dptcon) can be used to compute the condition number of $A$.

## 8Parallelism and Performance

f07jhf (dptrfs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07jhf (dptrfs) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $nr$. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The complex analogue of this routine is f07jvf (zptrfs).

## 10Example

This example solves the equations
 $AX=B ,$
where $A$ is the symmetric positive definite tridiagonal matrix
 $A = 4.0 -2.0 0 0 0 -2.0 10.0 -6.0 0 0 0 -6.0 29.0 15.0 0 0 0 15.0 25.0 8.0 0 0 0 8.0 5.0 and B = 6.0 10.0 9.0 4.0 2.0 9.0 14.0 65.0 7.0 23.0 .$
Estimates for the backward errors and forward errors are also output.

### 10.1Program Text

Program Text (f07jhfe.f90)

### 10.2Program Data

Program Data (f07jhfe.d)

### 10.3Program Results

Program Results (f07jhfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017